Geometry of Gauge Fields: Michal Kosztolowicz June 15, 2010
Geometry of Gauge Fields: Michal Kosztolowicz June 15, 2010
Geometry of Gauge Fields: Michal Kosztolowicz June 15, 2010
Michal Kosztolowicz
June 15, 2010
Abstract
The aim of this article is to give a mathematical formulation of Yang-
Mills gauge theory. Formalism of vector bundles and interpretation of a
field as a section of a vector bundle will be introduced. Using this language
notion of gauge transformation will be given and finally a connection on
a vector bundle will be defined, an object to be recognised as a gauge
potential.
1 Introduction
A gauge theory is a field theory in which the lagrangian is invariant under a
group of local transformations. Starting with the lagrangian which invariance
is only global, we can achieve a local invariance at the expense of introducing a
gauge field Aµ . It is used in a redefinition of a partial derivative inserted in the
lagrangian:
∂µ −→ Dµ = ∂µ + Aµ
Transformation rules of this new field are given as follows:
2 Vector bundles
Definition 1 A vector bundle is a quadruple (B, M, π, F), where B, M- smooth
manifolds called total space and base respectively, F - n dimensional vector space
called standard fibre and π : B → M is an onto map called projection, such that
following condition is satisfied:
Let {Uα } be the open covering of M. Then, for every α there exists a
diffeomorphism
tα : π −1 (Uα ) −→ Uα × F
1
such that its restriction tα,p ≡ tα |p : Bp ≡ π −1 (p) −→ {p} × F is a linear
isomorphism. This diffeomorphism (together with a set Uα ) is called a local
trivialization of a bundle.
That means that a section is a function that for every point p from the base
manifold picks out a single vector from a fibre π −1 (p). Shortly speaking, section
of a vector bundle is a vector field over a base manifold.
There always exists a global section of a vector bundle (that is defined on the
whole M, not only on a certain open subset). For example, one can consider
the zero section - it maps every point into zero vector and is independent of
2
local trivialization, since zero vector is always mapped into zero vector by an
isomorphism.
In a natural way there can be an addition and a multiplication by a function
from C ∞ (M) introduced in the set Γ(B). We define:
(f s)(p) ≡ f (p)s(p)
for arbitrary s, s0 ∈ Γ(B) and f ∈ C ∞ (M ). Using above it makes sense to talk
about a linear dependence of sections:
Definition 4 We say that e1 , e2 . . . en ∈ Γ(B) form a basis of sections, if every
s ∈ Γ(B) can be written as
s = si ei
where si are appropriate functions from C ∞ (M).
A basis of sections can be only defined for trivial bundles. That means
that for an arbitrary vector bundle we can have a basis only localy (for local
trivializations).
4 Gauge transformation
Using above introduced formalism notion of a gauge transformation can be
defined. For this purpose one needs a section of an endomorphism bundle which
is the same as just mentioned tensor bundle B ∗ ⊗ B, due to the isomorphism
V∗ ⊗ V ∼= End(V), where V is an arbitrary, finite dimensional vector space. Let
T ∈ Γ(B ∗ ⊗ B). It can be shown that T (p) ∈ Bp∗ ⊗ Bp ∼ = End(Bp ), so it is a
linear map acting on vectors from Bp . Thus, given a section s ∈ Γ(B) T defines
a new section T (s) ∈ Γ(B) as follows:
T (s)(p) = T (p)s(p)
After this introduction a definition can be given:
Definition 5 We say, that T (p) ∈ End(Bp ) lives in G, if it is of the form
v → gv for some g ∈ G and v ∈ Bp . If T (p) lives in G for every p, then we call
T a gauge transformation.
3
5 Connection on a vector bundle
5.1 Definition and transformation rules
Definition 6 Connection on a vector bundle is a map
D : Γ(B) −→ Γ(T ∗ M ⊗ B)
D(f s) = df ⊗ s + f Ds
ω11 ω1n
···
ω = ... ..
.
ωn1 ··· ωnn
Then we end up with a simple realtion
DS = ω ⊗ S (1)
4
properties of the connection matrix. For this reason consider change of a basis
of sections given by a matrix A
S 0 = AS
Putting it into (1) one finds
DS 0 = D(AS) = dA ⊗ S + ADS = dA ⊗ S + A(ω ⊗ S) = (dA + Aω) ⊗ S =
= (dAA−1 + AωA−1 ) ⊗ S 0 = ω 0 ⊗ S 0
An improtant formula for a transformation of connection is then derived:
ω 0 = dAA−1 + AωA−1
One immediately notices that this is precisely a transformation rule of a
potential of a gauge field. That is why those objects are identified.
DX s ≡< X, Ds >
If X = ∂µ , we denote D∂µ ≡ Dµ . We can rewrite Dµ s in terms of basis sections
eα of Γ(B) as follows:
Dµ s =< ∂µ , Ds >=< ∂µ , D(sα eα ) >=< ∂µ , dsα ⊗ eα + sα Deα >=
=< ∂µ , dsα ⊗ eα > + < ∂µ , sα ωαβ ⊗ eβ >= dsα (∂µ )eα + sα ωαβ (∂µ )eβ =
= ∂µ sα eα + sα ωαβ µ eβ = (∂µ sα + sβ ωβα µ )eα
As a result, we arrive at a desired formula for a coordinate representation of the
covariant derivative:
(Dµ s)α = ∂µ sα + sβ ωβα µ
References
[1] J. Baez, J.P. Muniain: Gauge fields, knots and gravity, World Scientific
Publishing Co. Pte. Ltd., Singapore 1994.
[2] L. Fatibene, M. Francaviglia: Natural and gauge natural formalism for
classical field theories. A geometric perspective including spinors and gauge
theories, Kluwer Academic Publishers, Dordrecht/Boston/London 2003.
[3] S.S. Chern, W.H.Chen, K.S. Lam: Lectures on differential geometry,
World Scientific Publishing Co. Pte. Ltd., Singapore 2000.
[4] Ch. N. Yang: Selected papers (1945-1980) with commentary, World Scien-
tific Publishing Co. Pte. Ltd., Singapore 2005.